Active Orders for Matroid Bases

We introduce three orderings of the basis set of an ordered matroid, defined in terms of basis activities. We show that the active orders have lattice structures. This property contains both the neighboring property of the lexicographic ordering of bases, and the maximality property of the product ordering, a form of the Greedy Algorithm Theorem. The active orders are closely related to the Tutte polynomial and the Orlik–Solomon algebra of a matroid. In particular, we show that, in a simple ordered matroid, a basis is the join in the external lattice of its components in the NBC basis of the Orlik–Solomon algebra of the matroid.